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Matthias Koeppe
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src/sage/rings/localization.py: Change # needs sage.rings.finite_rings to more precise sage.libs.pari; use more block tags
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‎src/sage/rings/localization.py

+29-23
Original file line numberDiff line numberDiff line change
@@ -77,8 +77,9 @@
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Obtain specializations in positive characteristic::
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sage: # needs sage.libs.pari sage.modules
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sage: Fp = GF(17)
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sage: f = L.hom((3,5,7,11), codomain=Fp); f # needs sage.libs.pari
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sage: f = L.hom((3,5,7,11), codomain=Fp); f
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Ring morphism:
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From: Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring localized at
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(q, q + 1, u2, u1, u1 - u2, u0, u0 - u2, u0 - u1, u2*q - u1, u2*q - u0,
@@ -88,17 +89,17 @@
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u1 |--> 5
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u2 |--> 7
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q |--> 11
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sage: mFp1 = matrix({k: f(v) for k, v in m1.dict().items()}); mFp1 # needs sage.libs.pari sage.modules
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sage: mFp1 = matrix({k: f(v) for k, v in m1.dict().items()}); mFp1
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[5 0 0]
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[0 3 0]
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[0 0 3]
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sage: mFp1.base_ring() # needs sage.libs.pari sage.modules
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sage: mFp1.base_ring()
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Finite Field of size 17
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sage: mFp2 = matrix({k: f(v) for k, v in m2.dict().items()}); mFp2 # needs sage.libs.pari sage.modules
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sage: mFp2 = matrix({k: f(v) for k, v in m2.dict().items()}); mFp2
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[ 2 3 0]
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[ 9 8 0]
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[ 0 0 16]
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sage: mFp3 = matrix({k: f(v) for k, v in m3.dict().items()}); mFp3 # needs sage.libs.pari sage.modules
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sage: mFp3 = matrix({k: f(v) for k, v in m3.dict().items()}); mFp3
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[16 0 0]
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[ 0 4 5]
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[ 0 7 6]
@@ -116,25 +117,28 @@
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u1 |--> 5
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u2 |--> 7
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q |--> 11
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sage: mQ1 = matrix({k: fQ(v) for k, v in m1.dict().items()}); mQ1 # needs sage.modules sage.rings.finite_rings
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sage: # needs sage.libs.pari sage.modules sage.rings.finite_rings
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sage: mQ1 = matrix({k: fQ(v) for k, v in m1.dict().items()}); mQ1
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[5 0 0]
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[0 3 0]
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[0 0 3]
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sage: mQ1.base_ring() # needs sage.modules sage.rings.finite_rings
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sage: mQ1.base_ring()
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Rational Field
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sage: mQ2 = matrix({k: fQ(v) for k, v in m2.dict().items()}); mQ2 # needs sage.modules sage.rings.finite_rings
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sage: mQ2 = matrix({k: fQ(v) for k, v in m2.dict().items()}); mQ2
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[-15 -14 0]
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[ 26 25 0]
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[ 0 0 -1]
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sage: mQ3 = matrix({k: fQ(v) for k, v in m3.dict().items()}); mQ3 # needs sage.modules sage.rings.finite_rings
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sage: mQ3 = matrix({k: fQ(v) for k, v in m3.dict().items()}); mQ3
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[ -1 0 0]
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[ 0 -15/26 11/26]
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[ 0 301/26 275/26]
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sage: # needs sage.libs.pari sage.libs.singular
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sage: S.<x, y, z, t> = QQ[]
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sage: T = S.quo(x + y + z)
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sage: F = T.fraction_field() # needs sage.libs.pari sage.libs.singular
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sage: fF = L.hom((x, y, z, t), codomain=F); fF # needs sage.libs.pari sage.libs.singular
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sage: F = T.fraction_field()
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sage: fF = L.hom((x, y, z, t), codomain=F); fF
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Ring morphism:
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From: Multivariate Polynomial Ring in u0, u1, u2, q over Integer Ring
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localized at (q, q + 1, u2, u1, u1 - u2, u0, u0 - u2, u0 - u1,
@@ -145,11 +149,11 @@
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u1 |--> ybar
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u2 |--> zbar
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q |--> tbar
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sage: mF1 = matrix({k: fF(v) for k, v in m1.dict().items()}); mF1 # needs sage.libs.pari sage.libs.singular sage.modules
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sage: mF1 = matrix({k: fF(v) for k, v in m1.dict().items()}); mF1 # needs sage.modules
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[ ybar 0 0]
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[ 0 -ybar - zbar 0]
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[ 0 0 -ybar - zbar]
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sage: mF1.base_ring() == F # needs sage.libs.pari sage.libs.singular sage.modules
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sage: mF1.base_ring() == F # needs sage.modules
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True
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TESTS::
@@ -211,15 +215,16 @@ def normalize_extra_units(base_ring, add_units, warning=True):
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....: [3*x, z*y**2, 2*z, 18*(x*y*z)**2, x*z, 6*x*z, 5])
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[z, y, x]
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sage: # needs sage.libs.singular
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sage: R.<x, y> = ZZ[]
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sage: Q.<a, b> = R.quo(x**2 - 5) # needs sage.libs.singular
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sage: p = b**2 - 5 # needs sage.libs.singular
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sage: p == (b-a)*(b+a) # needs sage.libs.singular
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sage: Q.<a, b> = R.quo(x**2 - 5)
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sage: p = b**2 - 5
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sage: p == (b-a)*(b+a)
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True
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sage: normalize_extra_units(Q, [p]) # needs sage.libs.pari sage.libs.singular
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sage: normalize_extra_units(Q, [p]) # needs sage.libs.pari
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doctest:...: UserWarning: Localization may not be represented uniquely
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[b^2 - 5]
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sage: normalize_extra_units(Q, [p], warning=False) # needs sage.libs.pari sage.libs.singular
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sage: normalize_extra_units(Q, [p], warning=False) # needs sage.libs.pari
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[b^2 - 5]
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"""
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# convert to base ring
@@ -256,7 +261,7 @@ class LocalizationElement(IntegralDomainElement):
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EXAMPLES::
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sage: # needs sage.rings.finite_rings
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sage: # needs sage.libs.pari
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sage: from sage.rings.localization import LocalizationElement
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sage: P.<x,y,z> = GF(5)[]
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sage: L = P.localization((x, y*z - x))
@@ -716,7 +721,7 @@ def _repr_(self):
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EXAMPLES::
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sage: R.<a> = GF(3)[]
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sage: Localization(R, a**2 - 1) # needs sage.rings.finite_rings
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sage: Localization(R, a**2 - 1) # needs sage.libs.pari
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Univariate Polynomial Ring in a over Finite Field of size 3
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localized at (a + 1, a + 2)
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"""
@@ -967,7 +972,7 @@ def fraction_field(self):
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EXAMPLES::
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sage: # needs sage.rings.finite_rings
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sage: # needs sage.libs.pari
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sage: R.<a> = GF(5)[]
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sage: L = Localization(R, (a**2 - 3, a))
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sage: L.fraction_field()
@@ -983,9 +988,10 @@ def characteristic(self):
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EXAMPLES::
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sage: # needs sage.libs.pari
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sage: R.<a> = GF(5)[]
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sage: L = R.localization((a**2 - 3, a)) # needs sage.rings.finite_rings
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sage: L.characteristic() # needs sage.rings.finite_rings
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sage: L = R.localization((a**2 - 3, a))
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sage: L.characteristic()
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5
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"""
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return self.base_ring().characteristic()

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